Review of Propositional Logic, Satisfiability, GSAT, WSAT

Report
Review of Propositional Logic
Syntax
• Propositional variables: x1, x2, …, xn (or may use other
symbols such as a, b, c, p, q, r).
• Logical connectives: ^ (and), v (or), ¬ (not, also sometimes
represented as ~ ), ← (left implication), → (right
implication).
• A proposition (propositional formula) is
–
–
–
–
any propositional variable
¬ α where α is any proposition
α ^ β where α and β are any propositions
likewise for all other logical connectives
• ¬ has higher precedence than binary connectives;
parentheses are used elsewhere to clarify precedence
Review of Propositional Logic
Semantics
• A truth assignment is a mapping of each
propositional variable to True (1) or False
(0)
• The “meaning” of any proposition is the set
of truth assignments that satisfy it, i.e. make
it true… these assignments are its models
Satisfaction
• A truth assignment A satisfies a proposition iff (if
and only if):
– The proposition is a variable that A maps to True
– The proposition is of the form ¬ α and A does not
satisfy α
– The proposition is of the form
• α ^ β and A satisfies α and β
• α v β and A satisfies α or β
• α → β and if A satisfies α then it satisfies β (i.e., A satisfies β or
does not satisfy α, possibly both)
Truth Tables
• A truth table for a propositional formula
shows all truth assignments for the variables
in the formula, and the truth of the formula
under each assignment.
• An example: p q r (pvq)→r
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
1
0
1
0
1
0
1
More Definitions
• Two propositions are equivalent iff they
have the same models, and hence the same
truth tables
• One proposition entails a second iff every
model of the first is a model of the second
Some Equivalences
•
•
•
•
¬¬α=α
(α → β) = (¬α v β)
DeMorgan’s Law: ¬(α ^ β) = (¬α v ¬β)
Distributivity of ^ over v: α ^ (β v γ) = (α ^ β) v (α
^ γ)
• Can show analogous equivalences for
distributivity of v over ^ and for commutativity
and associativity of ^ and v
• Exercise: Show these using truth tables and also
show you can swap ^ and v in DeMorgan’s Law.
Clausal Form or Conjunctive
Normal Form (CNF)
• A literal is a propositional variable or its
negation
• A clause is a disjunction of literals
• A CNF formula is a conjunction of clauses
• Can convert any formula into CNF form by
rewriting using the preceding equivalences
Review of Satisfiability
• A problem instance is a Boolean
conjunctive normal form (CNF) formula,
that is, a conjunction of propositional
clauses, over some set X1,…,Xn of
propositions.
• Goal is to find an assignment to the
propositions (variables) that satisfies the
CNF formula.
Satisfiability Review (Continued)
• Satisfiability is important for several
reasons, including
– It is at the foundation of NP-completeness
– It’s the canonical example of constraint
satisfaction problems (CSPs)
– Many interesting tasks, including planning
tasks, can be encoded as satisfiability problems.
• Broadly speaking, CSPs grow easier with
Satisfiability (Continued)
• (Continued)… more variables but harder
with more constraints. In the case of
satisfiability, each clause is a constraint.
• Kautz, Levesque, Mitchell, and Selman
showed that the critical measure of hardness
of satisfiability is the fraction of the number
of clauses over the number of variables.
For a large fraction, it’s almost always easy
Satisfiability (Continued)
• (Continued)… to answer “no” quickly, and
for a small fraction it’s almost always easy
to answer “yes” quickly. There’s a
relatively slim phase transition area in
between these extremes where most of the
hard problems are located.
• GSAT and WSAT were created (by subsets
of the preceding authors) to address these.
GSAT
• Input: CNF formula and integers Max_flips
(e.g. 100) and Max_climbs (e.g. 20).
• Output: Yes (satisfiable) or No (couldn’t
find a satisfying assignment). Might also
output the best assignment found.
• Assignments are scored by the number of
clauses they satisfy.
• GSAT performs a beam search with random
restarts and beam size set to 1.
GSAT Algorithm
• For i from 1 to Max_Climbs:
– Randomly draw a truth assignment over the
variables in the CNF formula (e.g. flip a coin
for each variable to decide whether to make it 0
or 1 -- in practice, use pseudo-random number).
If assignment satisfies formula, return “Yes”.
– For j from 1 to Max_Flips:
• For each variable, calculate the score of the truth
assignment that results when we flip the value of
GSAT Algorithm (Continued)
• (Continued)… that variable. Make the flip that
yields the highest score (need not be greater than or
equal to the score of the previous assignment). If
the new assignment satisfies the formula, return
“Yes”.
• Return “No” (no satisfying assignment
found, although one might still exist).
Key Points about GSAT
• Cannot tell us a formula is unsatisfiable (but
we can just run propositional resolution in
parallel).
• Random re-starts help us find multiple local
optima -- the hope is that one will be global.
• “Sideways” (or even “downward”) moves
help us get off a plateau -- can bounce us off
a local optimum. Significant practical
advance over standard greedy approach.
WalkSAT (WSAT)
• To further get around the problems of local
optima, we can occasionally choose to make
a random flip rather than a GSAT flip (as in
a random walk). WSAT differs from GSAT
as follows:
– One additional input: a probability p of a
random move at any step.
– A random move will involve flipping a variable
in some unsatisfied clause (efficacious flip).
WSAT (Continued)
– (Continued)… This change will satisfy some
currently unsatisfied clause (even if the net
result of the flip is a decrease in score).
– For each move, draw a pseudo-random number
between 0 and 1. If less than p, make a random
move; otherwise, make a GSAT move.
• WSAT outperforms GSAT, Simulated
Annealing, and a Genetic Algorithm on
random trials.

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